Three philosophical
views of mathematics are widely regarded as the ‘classic’ ones. Logicism holds that mathematics is reducible
to principles of pure logic. Intuitionism holds that mathematics is concerned with mental constructions
and defends a revision of classical mathematics and logic. Finally, formalism is the view that much or all
of mathematics is devoid of content and a purely formal study of strings of
mathematical language.

In recent
decades, some new views have entered the fray. An important newer arrival is structuralism, which holds that
mathematics is the study of abstract structures. A non-eliminative version of structuralism holds that there exist such things as
abstract structures, whereas an eliminative
version tries to make do with concrete objects variously structured. Nominalism denies that there are any
abstract mathematical objects and tries to reconstruct classical mathematics accordingly. Fictionalism
is based on the idea that, although most mathematical theorems are literally
false, there is a non-literal (or fictional) sense in which assertions of them
nevertheless count as correct. Mathematical
naturalism urges that mathematics be taken as a sui generis discipline in
good scientific standing.